Bous eld Localization Functors and Hopkins Chromatic

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2 MARK HOVEY, The major result of this paper is that finite torsion spectra are local with. respect to any infinite wedge of Morava K theories i K ni This has several. interesting corollaries For example it implies that there are no maps from the. Johnson Wilson spectra BP hni to a finite spectrum It also implies that if E is. a ring spectrum which detects all finite spectra so that E X 6 0 if X finite. then LE X is either X or Xp the p completion of X for finite X This in turn. implies that the only smashing localization which detects all finite complexes is. the identity functor, In order to prove that finite torsion spectra are i K ni local I show that. BPp is a wedge summand of i LK ni BPp This is saying that one does not. have to reassemble the chromatic pieces of BPp into an inverse limit to recover. the homotopy theory of BPp This result is a BP analogue of the chromatic. splitting conjecture of Hopkins I will describe this conjecture in Section 4 but. for now suffice it to say that the conjecture is that Ln 1 Xp is a wedge summand. in Ln 1 LK n Xp The chromatic splitting conjecture is actually stronger than. that for it also explains how this splitting occurs but most of the corollaries I. draw from the chromatic splitting conjecture only need the splitting itself One. corollary of the chromatic splitting conjecture would be that for finite X Xp is. a wedge summand of i LK ni Xp explaining how my result is a BP analogue. of the chromatic splitting conjecture I do not know if Ln 1 BPp is a wedge. summand of Ln 1 LK n BPp, In the first two sections of this paper I describe some other results about. Bousfield localization functors this time with respect to spectra E which kill a. finite spectrum The pedigree of these results is somewhat confusing Almost. all of the results in Sections 1 and 2 have been known to Hopkins for some. time Others may have known them as well but they have not appeared in. print before I feel that they warrant a larger audience In addition I discovered. many of these results independently and there are a couple of new results as. well For example I show that LK n is a minimal localization functor That is. if the natural transformation X LE X factors through LK n X then LE X is. either the zero functor or is LK n X itself I also provide some new examples of. smashing localizations, The last section of the paper discusses the consequences of the chromatic. splitting conjecture on the homotopy groups of Ln S 0 We show that given the. chromatic splitting conjecture the divisible summands in Ln S 0 for n 1 can. be determined There are 3n 1 of them with 2n 1 of them occuring in dimension. 2n and the others spread out from dimension 2n 1 to dimension n2 1. This therefore explains part of the Shimomura Yabe calculation of L2 S 0 for. p 3 in Sh Y, This paper is written in the homotopy category of p local spectra In parti.
cular is equality in the homotopy category namely homotopy equivalence. Similarly I will often write that a map or spectrum is 0 by which I mean that. it is null homotopic or contractible,CHROMATIC SPLITTING CONJECTURE 3. I would like to thank Mike Hopkins for sharing his ideas so freely I thank. Hal Sadofsky for hundreds of discussions on matters related to this paper I. also thank David Johnson for helpful discussions and Paul Eakin and Avinash. Sathaye for convincing me that my original ideas about infinite abelian groups. were too naive,1 Spectra with finite acyclics, Before describing the results of this section I need to recall the definition of. the Bousfield class of a spectrum Bou79, Definition 1 1 Two spectra E and F are Bousfield equivalent if given any. spectrum X,E X 0 if and only if F X 0, Denote the equivalence class of E by hEi Define hEi hF i if and only if. F X 0 implies E X 0 Define,hEi hF i hE F i,hEi hF i hE F i.
There is a minimal Bousfield class h i which we will often denote by 0 and. a maximal Bousfield class hS 0 i I remind the reader that it is perfectly possible. to have hEi hF i while nonetheless hE F i 0, In this section we investigate Bousfield classes of spectra E which have finite. acyclics i e there is some finite X with E X 0 Highlights of this section. include the minimality of the Bousfield class of K n Corollary 1 7 and the new. examples of smashing localizations Proposition 1 5 We also show that every. BP module spectrum with finite acyclics has the Bousfield class of a wedge of. Morava K theories and that a vn periodic Landweber exact spectrum has the. same Bousfield class as E n, First we need to recall some corollaries of the nilpotence theorem DHS HS. Recall that a finite spectrum X has type n if K i X 0 for i n but. K n X 6 0 Every finite spectrum is of some finite type and the periodicity. theorem of J Smith written up in Rav92 says that there is a spectrum of type. n for all n Let Cn denote the class of all finite spectra of type at least n Then. HS any nonempty collection of finite spectra that is closed under cofibrations. and retracts is some Cn, Lemma 1 2 Hopkins Smith All finite spectra of type n have the same. Bousfield class which we denote F n, Proof This is an easy application of the nilpotence theorem Given an X. of type n let C consist of all finite spectra Y such that hY i hXi It is easy to. see that C is closed under retracts cofibrations and suspensions so must be a. 4 MARK HOVEY, Ck for some k Since X C C Cn In particular if Y is type n hY i hXi.
Interchanging X and Y completes the proof, A spectrum X in Cn has a vn self map that is a map inducing an isomorphism. on K n X HS Any two such become equal upon iterating them enough. times so that there is a well defined telescope T eln X T eln is actually an. exact functor on the category of finite spectra with vn self maps This follows. from the fact that a map between two such finite spectra will commute with. large enough iterates of the vn self maps By following a similar line of proof as. in the above lemma we get, Lemma 1 3 The telescopes of finite spectra of type n all have the same Bous. field class which we denote T el n, This lemma was also known to Hopkins and Smith and it appears in MS. Recall the lemma of Rav84 if f is a self map of X and T el X is its. telescope and Y its cofibre then hXi hT el X i hY i Applying this repeatedly. using vn self maps we get,1 hS 0 i hT el 0 i hT el n i hF n 1 i. This decomposition is the key to most of our results in this section Note that it. is orthogonal in the sense that T el m T el n T el m F n 0 if m n. This is proven in MS,Given any spectrum E let,FA E X X is finite and E X 0.
In this section we will discuss spectra which have finite acyclics so that we. assume FA E 6 It is easy to see that FA E is closed under cofibrations and. retracts so it must be Cn 1 for some n We then have the following observation. Lemma 1 4 If FA E Cn 1 then,hEi hE T el 0 i hE T el n i. In particular hT el 0 T el n i is the largest Bousfield class with finite acy. clics Cn 1 and therefore localization with respect to it denoted Lfn is smashing. Proof Just smash equation 1 with E To see Lfn is smashing note that. for any spectrum E FA LE S 0 FA E Thus,hLfn S 0 i hT el 0 T el n i. This implies by Prop 1 27 of Rav84 that Lfn is smashing. CHROMATIC SPLITTING CONJECTURE 5, Lfn has been investigated by many authors Bou92 MS Mil Rav92a All. of them noticed that it is smashing though I think this is the most transparent. proof The telescope conjecture is usually stated as saying that if X is type n. then LK n X T el X This is equivalent to hT el n i hK n i and also to. Lfn Ln For details see MS or one of the other cited papers above This. conjecture is now known to be false for n 2 Rav92b MRS and is presumed. to be false for larger n as well, As an amusing example of what the failure of the telescope conjecture means. we include the following proposition,Proposition 1 5 Localization with respect to.
T el 0 T el m K m 1 K n,is smashing, Proof Call this localization functor Lm n We have that. hT el 0 T el m i hLm n S 0 i hT el 0 T el n i, We need to show that hLm n S 0 T el i i hK i i if m i n Note that. hLm n S 0 T el i i hLm n S 0 F i T el i i hLm n F i T el i i. But F i is T el 0 T el m acyclic so Lm n F i Ln F i Since Ln is. smashing Rav92 hLn F i i hK i K n i and the result follows. It is an old problem of Bousfield s to classify all smashing localization functors. We address another part of this problem in Section 3. To measure the extent to which the telescope conjecture fails note that there. is a natural map Lfn X Ln X Let An X be the fibre of this map Note that. if X is type n this is also the fiber of the map T el X LK n X for then. Lfn X T el X MS and Ln X LK n X, Proposition 1 6 If X is finite and type n the Bousfield class of An X does. not depend on X We denote it A n hT el n i hK n i hA n i and A n. K m 0 for all m, Proof First note that because Lfn and Ln are both smashing see Rav92. for Ln so is An That is An X An S 0 X for all X In particular if X and. Y are Bousfield equivalent so are An X and An Y This shows that hA n i is. well defined The map Lfn X Ln X is an isomorphism on K m homology for. all m and also on BP homology as we will see below so A n K m 0. for all m If X is type n hT el X i hT el n i and hLK n Xi hK n i and it. follows that hT el n i hK n i hA n i, A n behaves very much like Mn X the nth monochromatic layer which is.
the fiber of Ln X Ln 1 X In particular we have that A n A n is homotopy. equivalent to A n and LA n A n is homotopy equivalent to LA n S 0. 6 MARK HOVEY, Corollary 1 7 hK n i is a minimal Bousfield class That is if hEi. hK n i then E is null,Proof Suppose hEi hK n i Then. hE T el m i hK n T el m i 0, if n 6 m Similarly hE F n 1 i hK n F n 1 i 0 Thus from. equation 1 we have that hEi hE T n i But also,hE A n i hK n A n i 0. so by the preceeding proposition we have hEi hE K n i Since K n is a. field spectrum E K n is a wedge of suspensions of K n so there are only. two possibilities for hE K n i 0 or hK n i, Note that the corresponding result is not true for the other field spectrum.
HFp Indeed in the proof of Theorem 2 2 of Rav84 Ravenel shows that. hY i hHFp i where Y denotes the Brown Comanetz dual of BP M p. A similar argument to the above shows that if hEi is less than or equal to. some finite wedge of Morava K theories then E must be Bousfield equivalent to. a finite wedge of Morava K theories This says in particular that the chromatic. tower is unrefineable There is no localization functor LE that fits between Ln. In the light of this result and the failure of the telescope conjecture one might. ask if A n is also a minimal Bousfield class This would say that the telescope. conjecture is not so badly wrong I think this is likely to be true but since I. have no data I will not be so bold as to conjecture it. The following theorem will show that the telescope conjecture is true after. smashing with BP This has been known to Hopkins Ravenel and probably. others though it has not appeared before First we need a lemma. Lemma 1 8 Suppose R is a ring spectrum and the unit map S 0 R factors. through some spectrum E Then hEi hRi,Proof Since R is a ring spectrum the composite. R S0 R R R R, is the identity Since factors through E the identity map on R factors through. E R So if E Z is null so is R Z, Recall that P n is a BP module spectrum whose homotopy is. P n BP p v1 vn 1, The first part of the following theorem is Ravenel s theorem 2 1 g in Rav84. We reprove it so as to make the second part clearer. CHROMATIC SPLITTING CONJECTURE 7,Theorem 1 9,hBP F n i hP n i.
hBP T el n i hK n i,Proof If there were a spectrum V n 1 with. BP V n 1 BP p v1 vn 1, it would be type n and we would have BP V n 1 P n so the result would. be obvious In general there are not such spectra but there are appropriate. substitutes M pi0 v1i1 vn 1 constructed by Devinatz in Dev These exist. for sufficiently large i0 i1 in 1 they are finite of type n and they have. the evident BP homology Furthermore BP M pi0 v1i1 vn 1 can be. constructed from P n using cofibre sequences in the same way that the mod. pn Moore space can be constructed from the mod p Moore space Therefore. hBP M pi0 v1i1 vn 1, Note that there is a natural map of BP module spectra. BP M pi0 v1i1 vn 1, The unit map S 0 P n of the ring spectrum P n factors through this map. so by the proceeding lemma,hBP M pi0 v1i1 vn 1, It can actually be shown using a variant of the Landweber exact functor theo.
rem and Lemma 2 13 of Rav84 that P n is a module spectrum over BP. M pi0 v1i1 vn 1 but we do not need this, To see that hBP T el n i hK n i we proceed similarly A vn self map on. M pi0 v1i1 vn 1 induces multiplication by a power of vn on BP homology. vn 1 BP M pi0 v1i1 vn 1,BP T el M pi0 v1i1 vn 1,The maps that build BP M pi0 v1i1 vn 1 n 1. from P n by cofibre sequences, can all be chosen to be BP module maps Thus they will also build vn 1 BP. from vn 1 P n Thus,M pi0 v1i1 vn 1,hBP T el n i hvn 1 P n i hK n i. The latter equality comes from Theorem 2 1 of Rav84. The unit map of vn 1 P n factors through vn 1 BP M pi0 v1i1 vn 1. so we also have hBP T el n i hK n i, Corollary 1 10 BP A n 0 so that the natural map Lfn X Ln X is.
a BP equivalence,8 MARK HOVEY,hBP A n i hBP T el n A n i hK n A n i 0. Corollary 1 11 Every BP module spectrum with finite acyclics is Bous. tors in stable homotopy theory All spectra will be p local for a prime pthroug hout this paper Recall that if Eis a spectrum a spectrum Xis E acyclic if E Xis null A spectrum is E local if every map from an E acyclic spectrum to it is null A map X Y is an E equivalence if it induces an isomorphism on E or equivalently if the bre is E acyclic In Bou79 Bous eld shows that there is a

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